Generalized multiplicative Sidon sets

Let us call a set of positive integers a multiplicative k  -Sidon set, if the equation a1a2…ak=b1b2…bka1a2…ak=b1b2…bk does not have a solution consisting of distinct elements of this set. Let Gk(n)Gk(n) denote the maximal size of a multiplicative k  -Sidon subset of {1,2,…,n}{1,2,…,n}. In this paper we prove that π(n)+π(n/2)+c1n2/3/(log⁡n)4/3≤G3(n)≤π(n)+π(n/2)+c2n2/3log⁡nlog⁡log⁡n for some constants c1,c2>0c1,c2>0. It is also shown that π(n)+n3/5/(log⁡n)6/5≤G4(n)≤π(n)+(10+ε)n2/3π(n)+n3/5/(log⁡n)6/5≤G4(n)≤π(n)+(10+ε)n2/3. Furthermore, for every k   the order of magnitude of Gk(n)Gk(n) is determined and an upper bound, similar to the previously mentioned ones, is given. This problem is related to a problem of Erdős–Sárközy–T. Sós and Győri: They examined how many elements of the set {1,2,…,n}{1,2,…,n} can be chosen in such a way that none of the 2k  -element products is a perfect square. The maximal size of such a subset is denoted by F2k(n)F2k(n). As a consequence of our upper estimates for Gk(n)Gk(n) the upper estimates for F2k(n)F2k(n) are strengthened because Gk(n)≥F2k(n)Gk(n)≥F2k(n). Moreover, by a new construction we also sharpen their lower bound for F8(n)F8(n).